A Seneta's Conjecture and the Williamson Transform

Considering slowly varying functions (SVF), %Seneta (۲۰۱۹) Seneta in ۲۰۱۹ conjectured the following implication, for \alpha\geq۱,\int_۰^x y^{\alpha-۱}(۱-F(y))dy\textrm{\ is SVF}\ \Rightarrow\ \int_{۰}^x y^{\alpha}dF(y)\textrm{\ is SVF, as x\to\infty,}where F(x) is a cumulative distribution function on [۰,\infty). By applying the Williamson transform, an extension of this conjecture is proved. Complementary results related to this transform and particular cases of this extended conjecture are discussed.

Keywords:

Regular variation , De Haan class , Truncated moments , Williamson transform